Perfect and nearly perfect sampling of work-conserving queues

Abstract

We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers.Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called "nearly" perfect sampling.Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method.Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems.

Sensitivity Analysis and Simulation of a Multiserver Queueing System with Mixed Service Time Distribution

“…However, we note that the steady-state exact sampling by regenerative simulation has several serious drawbacks. First, the average working time of the algorithm may be infinite [36], e.g., in a system with large number of servers (which indeed depends on the regenerative cycle length). This problem can be solved either by the coupling-from-the-past technique [35] (which, although, is rather technically tricky), or by non traditional regenerative techniques, such as the artificial regeneration [41] or regenerative envelopes [40].…”

“…We note that, although this approach allows to sample exactly from the steady-state distribution, the regeneration period in the dominated RA system can be very large in practice, and, thus, can lead to unacceptable long simulation. For further details on perfect sampling, see [15,[35][36][37]. Now we explain how to sample from the equilibrium distribution of a two-component mixture.…”

Perfect sampling of a single-server queue with periodic Poisson arrivals

“…(4) Unfortunately the regenerative method has infinite expected runtime [5,20]. Thus some runs may take so long that a practitioner would abort them, introducing a bias into the simulation.…”